Interpolating with outer functions

TL;DR

This paper investigates conditions under which interpolation problems in complex analysis can be solved with outer functions, refining existing results and exploring applications to Toeplitz operators on model spaces.

Contribution

It characterizes when interpolating functions can be outer functions, extending classical theorems and applying findings to operator theory on model spaces.

Findings

Conditions for outer function interpolation established

Refinement of McCarthy's result on interpolation

Analysis of co-analytic Toeplitz operators' common range

Abstract

The classical theorems of Mittag-Leffler and Weierstrass show that when $\{\lambda_n\}$ is a sequence of distinct points in the open unit disk $\D$, with no accumulation points in $\D$, and $\{w_n\}$ is any sequence of complex numbers, there is an analytic function $\phi$ on $\D$ for which $\phi(\lambda_n) = w_n$. A celebrated theorem of Carleson \cite{MR117349} characterizes when, for a bounded sequence $\{w_n\}$, this interpolating problem can be solved with a bounded analytic function. A theorem of Earl \cite{MR284588} goes further and shows that when Carleson's condition is satisfied, the interpolating function $\phi$ can be a constant multiple of a Blaschke product. In this paper, we explore when the interpolating $\phi$ can be an outer function. We then use our results to refine a result of McCarthy \cite{MR1065054} and explore the common range of the co-analytic Toeplitz operators on a model space.